Derivation of Elongation in a Tapered Bar with Circular Cross-section under an Applied Force

 

Derivation of Elongation in a Tapered Bar with Circular Cross-section under an Applied Force

Elongation or deformation of a structural element under an applied load is a fundamental concept in the field of mechanics of materials. In this blog, we will derive the elongation of a tapered bar with a circular cross-section when subjected to an axial force $P$. A tapered bar is one whose diameter changes along its length. Understanding how such a bar deforms under a load is crucial for designing various engineering components such as shafts, rods, and columns.

Assumptions

1. The material of the bar is homogeneous and isotropic.
2. The deformation is within the elastic limit of the material (Hooke's law is applicable).
3. The taper is linear, meaning the diameter changes linearly from one end to the other.
4. The axial force $P$ is applied uniformly along the length of the bar.

Geometry of the Tapered Bar

Consider a tapered bar with the following characteristics:

• Length of the bar: $L$
• Diameter at the larger end: $D_1$
• Diameter at the smaller end: $D_2$

Since the bar is tapered linearly, the diameter $d(x)$ at a distance $x$ from the larger end is given by: $d(x) = D_1 - \left( \frac{D_1 - D_2}{L} \right) x$

Differential Element Analysis

To derive the elongation, consider an infinitesimally small element $dx$ at a distance $x$ from the larger end. The diameter of this element is $d(x)$. The cross-sectional area $A(x)$ of the element is: $A(x) = \frac{\pi}{4} [d(x)]^2$ Substituting $d(x)$: $A(x) = \frac{\pi}{4} \left[ D_1 - \left( \frac{D_1 - D_2}{L} \right) x \right]^2$

Elongation of the Differential Element

The elongation $d\delta$ of the differential element $dx$ under the axial force $P$ is given by Hooke's law: $d\delta = \frac{P \, dx}{A(x) \, E}$ where $E$ is the modulus of elasticity of the material.

Substituting $A(x)$ into the elongation equation: $d\delta = \frac{P \, dx}{\frac{\pi}{4} \left[ D_1 - \left( \frac{D_1 - D_2}{L} \right) x \right]^2 \, E}$ $d\delta = \frac{4P \, dx}{\pi E \left[ D_1 - \left( \frac{D_1 - D_2}{L} \right) x \right]^2}$

Total Elongation of the Bar

To find the total elongation $\delta$, integrate the differential elongation over the length of the bar from $x = 0$ to $x = L$: $\delta = \int_0^L \frac{4P \, dx}{\pi E \left[ D_1 - \left( \frac{D_1 - D_2}{L} \right) x \right]^2}$

Let $k = \frac{D_1 - D_2}{L}$. Then the integral becomes: $\delta = \frac{4P}{\pi E} \int_0^L \frac{dx}{\left( D_1 - kx \right)^2}$

To solve this integral, use the substitution $u = D_1 - kx$, hence $du = -k \, dx$ and $dx = -\frac{du}{k}$. The limits of integration change accordingly:

• When $x = 0$, $u = D_1$
• When $x = L$, $u = D_2$

The integral becomes: $\delta = \frac{4P}{\pi E} \int_{D_1}^{D_2} \frac{-\frac{du}{k}}{u^2}$ $\delta = \frac{4P}{\pi E k} \int_{D_1}^{D_2} \frac{du}{u^2}$ $\delta = \frac{4P}{\pi E k} \left[ -\frac{1}{u} \right]_{D_1}^{D_2}$ $\delta = \frac{4P}{\pi E k} \left( -\frac{1}{D_2} + \frac{1}{D_1} \right)$ $\delta = \frac{4P}{\pi E k} \left( \frac{1}{D_1} - \frac{1}{D_2} \right)$

Substitute $k = \frac{D_1 - D_2}{L}$: $\delta = \frac{4P L}{\pi E (D_1 - D_2)} \left( \frac{1}{D_1} - \frac{1}{D_2} \right)$

Simplification and Final Formula

Simplify the expression to get the final formula for the elongation $\delta$ of the tapered bar: $\delta = \frac{4P L}{\pi E} \left( \frac{1}{D_1 D_2} \right)$

Thus, the elongation of a tapered bar with a circular cross-section under an axial force $P$ is given by: $\delta = \frac{4P L}{\pi E D_1 D_2}$

Conclusion

In this blog, we derived the elongation of a tapered bar with a circular cross-section when subjected to an axial force. The key steps involved analyzing a differential element of the bar, applying Hooke's law, and integrating over the length of the bar to find the total elongation. This derivation is crucial for understanding how tapered structural elements deform under load, which is essential for designing safe and efficient engineering components.